Scattering theory for the Dirac equation of Hartree type and the semirelativistic Hartree equation

Consider a scattering problem for the Dirac equation with a nonlocal term including the Hartree type. We improve the condition of the potential term to show the existence of scattering operators for small initial data in the subcritical Sobolev spaces. Our proofs can be applied to the case of the semirelativistic Hartree equation, and lead to improvement of the condition of the potential.     

Stationary solutions for a superlinear Dirac equation

K. Guerlebeck In this paper, we consider the following nonlinear Dirac equation By applying the variant generalized weak linking theorem for strongly indefinite problem developed by Schechter and Zou, we prove the existence of nontrivial and ground state solutions for the aforementioned system under conditions weaker than those in Zhang et al. (Journal of Mathematical Physics, 2013). John Wiley & Sons, Ltd.     

Semiclassical approximation of the Dirac equation in a central field

We develop a semiclassical approximation of the Dirac equation in a central field with a Coulomb asymptotic behavior. We obtain relativistic semiclassical scattering phases, energy levels of hydrogen-like ions, and a semiclassical expression for the multiplicative constant in the asymptotic expansion of the wave function of the valence electron in a relativistic multiply charged ion, which plays an important role in quantum defect theory.     

Scattering of solitons for Dirac equation coupled to a particle

We establish soliton-like asymptotics for finite energy solutions to the Dirac equation coupled to a relativistic particle. Any solution with initial state close to the solitary manifold, converges in long time limit to a sum of traveling wave and outgoing free wave. The convergence holds in global energy norm. The proof uses spectral theory and symplectic projection onto solitary manifold in the Hilbert phase space.     

WKB method for the Dirac equation with a scalar-vector coupling

We outline a recursive method for obtaining WKB expansions of solutions of the Dirac equation in an external centrally symmetric field with a scalar-vector Lorentz structure of the interaction potentials. We obtain semiclassical formulas for radial functions in the classically allowed and forbidden regions and find conditions for matching them in passing through the turning points. We generalize the Bohr-Sommerfeld quantization rule to the relativistic case where a spin-1/2 particle interacts simultaneously with a scalar and an electrostatic external field. We obtain a general expression in the semiclassical approximation for the width of quasistationary levels, which was earlier known only for barrier-type electrostatic potentials (the Gamow formula). We show that the obtained quantization rule exactly produces the energy spectrum for Coulomb- and oscillatory-type potentials. We use an example of the funnel potential to demonstrate that the proposed version of the WKB method not only extends the possibilities for studying the spectrum of energies and wave functions analytically but also ensures an appropriate accuracy of calculations even for states with n_r 1.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 1, pp. 83–111, April, 2005.     

Global strong solution to a nonlinear Dirac type equation in one dimension

This paper studies a class of nonlinear massless Dirac equations in one dimension, which include the equations for the massless Thirring model and the massless Gross–Neveu model. Under the assumptions of the initial data having small charge and being bounded, the global existence of the strong solution is established. The decay of the local charge is also proved.     

On a boundary value problem for a p‐Dirac equation

The p‐Laplace equation is a nonlinear generalization of the Laplace equation. This generalization is often used as a model problem for special types of nonlinearities. The p‐Laplace equation can be seen as a bridge between very general nonlinear equations and the linear Laplace equation. The aim of this paper is to solve the p‐Laplace equation for 1 < p < 2 and to find strong solutions. The idea is to apply a hypercomplex integral operator and spatial function theoretic methods to transform the p‐Laplace equation into the p‐Dirac equation. This equation will be solved iteratively by using a fixed‐point theorem. Applying operator‐theoretical methods for the p‐Dirac equation and p‐Laplace equation, the existence and uniqueness of solutions in certain Sobolev spaces will be proved. Copyright © 2016 John Wiley & Sons, Ltd.     

Bosonic symmetries of the massless Dirac equation

The results of spin 1 symmetries of massless Dirac equation [21] are proved completely in the space of 4-component Dirac spinors on the basis of unitary operator in this space connecting this equation with the Maxwell equations containing gradient-like sources. Nonlocal representations of conformal group are found, which generate the transformations leaving the massless Dirac equation being invariant. The Maxwell equations with gradient-like sources are proved to be invariant with respect to fermionic representations of Poincaré and conformal groups and to be the kind of Maxwell equations with maximally symmetrical properties. Brief consideration of an application of these equations in physics is discussed.     

Scattering data for the nonstationary Dirac equation

The existence of indeterminacy in the choice of scattering data for the auxiliary linear system for the Davey-Stewartson I-equation is noted. A connection is established between different scattering data and the corresponding conjugation matrix for the nonlocal Riemann problem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 164, pp. 170–175, 1987.     

The explicit solutions to the nonlinear Dirac equation and Dirac-Klein-Gordon equation

Abstract In [3] Dias and Figueira have reported that the square of the solution for the nonlinear Dirac equation satisfies the linear wave equation in one space dimension. So the aim of this paper is to proceed with their work and to clarify a structure of the nonlinear Dirac equation. The explicit solutions to the nonlinear Dirac equation and Dirac-Klein-Gordon equation are obtained. Keywords: Nonlinear Dirac equation, Dirac-Klein-Gordon equation, Pauli matrix Mathematics Subject Classification (2000): 35C05, 35L45     

Fundamental solution of the stationary Dirac equation with a linear potential

Theoretical and Mathematical Physics - We explicitly express the fundamental solution of the stationary two-dimensional massless Dirac equation with a constant electric field in terms of Fourier...     

Relative oscillation theory for Dirac operators

We develop relative oscillation theory for one-dimensional Dirac operators which, rather than measuring the spectrum of one single operator, measures the difference between the spectra of two different operators. This is done by replacing zeros of solutions of one operator by weighted zeros of Wronskians of solutions of two different operators. In particular, we show that a Sturm-type comparison theorem still holds in this situation and demonstrate how this can be used to investigate the number of eigenvalues in essential spectral gaps. Furthermore, the connection with Krein's spectral shift function is established. As an application we extend a result by K.M. Schmidt on the finiteness/infiniteness of the number of eigenvalues in essential spectral gaps of perturbed periodic Dirac operators.